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Mirrors > Home > ILE Home > Th. List > elfz2 | Unicode version |
Description: Membership in a finite set of sequential integers. We use the fact that an operation's value is empty outside of its domain to show and . (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.) |
Ref | Expression |
---|---|
elfz2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | anass 399 | . 2 | |
2 | df-3an 975 | . . 3 | |
3 | 2 | anbi1i 455 | . 2 |
4 | df-fz 9966 | . . . 4 | |
5 | 4 | elmpocl 6047 | . . 3 |
6 | simpl 108 | . . 3 | |
7 | elfz1 9970 | . . . 4 | |
8 | 3anass 977 | . . . . 5 | |
9 | ibar 299 | . . . . 5 | |
10 | 8, 9 | syl5bb 191 | . . . 4 |
11 | 7, 10 | bitrd 187 | . . 3 |
12 | 5, 6, 11 | pm5.21nii 699 | . 2 |
13 | 1, 3, 12 | 3bitr4ri 212 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 w3a 973 wcel 2141 crab 2452 class class class wbr 3989 (class class class)co 5853 cle 7955 cz 9212 cfz 9965 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-setind 4521 ax-cnex 7865 ax-resscn 7866 |
This theorem depends on definitions: df-bi 116 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-opab 4051 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-iota 5160 df-fun 5200 df-fv 5206 df-ov 5856 df-oprab 5857 df-mpo 5858 df-neg 8093 df-z 9213 df-fz 9966 |
This theorem is referenced by: elfz4 9974 elfzuzb 9975 uzsubsubfz 10003 fzmmmeqm 10014 fzpreddisj 10027 elfz1b 10046 fzp1nel 10060 elfz0ubfz0 10081 elfz0fzfz0 10082 fz0fzelfz0 10083 fz0fzdiffz0 10086 elfzmlbp 10088 fzind2 10195 iseqf1olemqcl 10442 iseqf1olemnab 10444 iseqf1olemab 10445 seq3f1olemqsumkj 10454 seq3f1olemqsumk 10455 summodclem2a 11344 fsum3 11350 fsum3cvg3 11359 fsumcl2lem 11361 fsumadd 11369 fsummulc2 11411 prodmodclem3 11538 prodmodclem2a 11539 fprodntrivap 11547 fprodeq0 11580 isprm5 12096 |
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